Hashing and finding closest points

Question 1

What is a dictionary data structure?

Answer 1

A dictionary is a data structure that supports MakeDictionary

Question 2

What is the universe U in the context of dictionaries?

Answer 2

The universe ( U ) is the set of all possible elements that could be part of the subset ( S ) maintained by the dictionary.

Question 3

Why is hashing used for dictionaries?

Answer 3

Hashing allows efficient dictionary operations by mapping a large universe ( U ) into a smaller hash table using a hash function.

Question 4

What is a hash function?

Answer 4

A hash function maps elements from the universe ( U ) to positions in a smaller array (hash table) to store and retrieve elements efficiently.

Question 5

What is a collision in a hash table?

Answer 5

A collision occurs when two different elements ( u ) and ( v ) are mapped to the same hash table position by the hash function.

Question 6

How does chaining resolve hash collisions?

Answer 6

In chaining

Question 7

What are the operations supported by hash tables?

Answer 7

Hash tables support Insert (add an element), Delete (remove an element), and Lookup (check membership and retrieve data).

Question 8

What makes a hash function ‘good’?

Answer 8

A good hash function minimizes collisions by distributing elements uniformly across the hash table.

Question 9

What is the probability of a collision in uniform random hashing?

Answer 9

With uniform random hashing, the probability that two random elements collide is exactly ( 1/n ), where ( n ) is the size of the hash table.

Question 10

What is a universal class of hash functions?

Answer 10

A universal class of hash functions ensures that for any two distinct elements ( u ) and ( v ), the probability that ( h(u) = h(v) ) is at most ( 1/n ).

Question 11

What is the advantage of universal hash functions?

Answer 11

Universal hash functions ensure efficient dictionary operations by reducing the likelihood of collisions, even in the worst case.

Question 12

How is a universal hash function constructed?

Answer 12

A universal hash function can be constructed using linear functions modulo a prime ( p ), such as ( h(x) = (\sum a_i x_i) \mod p ).

Question 13

Why is a prime ( p ) used in universal hashing?

Answer 13

Using a prime ( p ) ensures that modular arithmetic avoids divisibility issues, providing better collision guarantees.

Question 14

What is the complexity of dictionary operations using hashing?

Answer 14

With universal hashing, the expected complexity of Insert, Delete, and Lookup operations is ( O(1) ).

Question 15

How are hash tables used in the closest-pair algorithm?

Answer 15

Hash tables help divide the unit square into subsquares, allowing efficient Lookup of nearby points during the closest-pair algorithm.

Question 16

What is the expected time complexity of the randomized closest-pair algorithm?

Answer 16

The algorithm runs in ( O(n) ) time for processing points plus ( O(n) ) dictionary operations.

Question 17

Why is randomness used in the closest-pair algorithm?

Answer 17

Randomness is used to order points and in hash functions, ensuring efficiency and reducing worst-case scenarios.

Question 18

What are subsquares in the closest-pair algorithm?

Answer 18

Subsquares divide the unit square into regions of side length ( \delta/2 ), used to group points for efficient distance computation.

Question 19

What is the significance of the subsquares’ properties?

Answer 19

Subsquares ensure that only nearby points need to be checked for distances less than ( \delta ), reducing unnecessary computations.

Question 20

How does the closest-pair algorithm update its state when ( \delta ) changes?

Answer 20

When ( \delta ) decreases, a new dictionary is created with updated subsquares, and all points are reinserted.

Question 21

What is the relationship between hash table collisions and dictionary performance?

Answer 21

Fewer collisions lead to shorter chains in hash table entries, improving the efficiency of dictionary operations.

Question 22

How is a hash value computed efficiently in universal hashing?

Answer 22

Hash values are computed using linear functions with modular arithmetic, requiring ( O(\log N / \log n) ) operations for ( n ) elements.

Question 23

Why is universal hashing effective for dynamic dictionaries?

Answer 23

Universal hashing minimizes collisions across arbitrary input sets, ensuring efficient operations even with adversarial input.

Question 24

How does randomization in hashing improve performance?

Answer 24

Randomization spreads elements uniformly across the hash table, reducing the chance of clustering and long chains.

Question 25

What is the expected number of Insert operations in the closest-pair algorithm?

Answer 25

The expected number of Insert operations is ( O(n) ), as updates to the closest pair become less frequent over time.

Question 26

How do dictionary operations contribute to the overall complexity of the closest-pair algorithm?

Answer 26

Efficient dictionary operations ensure the total work remains ( O(n) ) in expectation, matching the algorithm’s computational efficiency.